A representation of twistors within geometric (Clifford) algebra

TL;DR

This paper extends the geometric algebra approach to twistors, providing a clearer geometric interpretation, simplifying the formalism, and connecting twistors to quantum systems within a conformal geometric framework.

Contribution

It introduces a new geometric algebra-based representation of twistors, clarifies their position dependence, and maps them to a 6-d conformal space for quantum system analysis.

Findings

Derived the spinor representation of the conformal group in geometric algebra.

Mapped twistors to 6-d conformal space as quantum states.

Simplified the twistor formalism using geometric algebra.

Abstract

In previous work by two of the present authors, twistors were re-interpreted as 4-d spinors with a position dependence within the formalism of geometric (Clifford) algebra. Here we extend that approach and justify the nature of the position dependence. We deduce the spinor representation of the restricted conformal group in geometric algebra, and use it to show that the position dependence is the result of the action of the translation operator in the conformal space on the 4-d spinor. We obtain the geometrical description of twistors through the conformal geometric algebra, and derive the Robinson congruence. This verifies our formalism. Furthermore, we show that this novel approach brings considerable simplifications to the twistor formalism, and new advantages. We map the twistor to the 6-d conformal space, and derive the simplest geometrical description of the twistor as an observable of a relativistic quantum system. The new 6-d twistor takes the r\^ole of the state for that system. In our new interpretation of twistors as 4-d spinors, we therefore only need to apply the machinery already known from quantum mechanics in the geometric algebra formalism, in order to recover the physical and geometrical properties of 1-valence twistors.